A module is called non-co-Hopfian if it admits a proper embedding into itself. In this talk I will discuss combinatorial structures and invariants associated with such modules, with particular focus on representations of generalized Weyl algebras.

A *-algebra is an algebra equipped with an involution. The Cayley–Dickson construction builds new *-algebras from a given one, generalizing the construction of the complex numbers from the real numbers. Classically, this uses an involution corresponding to complex conjugation - but is that the only choice? (pdf)

With motivation from noncommutative geometry, we want to know when there exists a *-algebra structure on path algebras of quivers. We consider quivers with anti-involutions and show that a path algebra admits a *-algebra structure if and only if there exists an anti-involution on its underlying quiver. We finish the talk with a discussion of future investigations. (pdf)

Rigidity theory studies when a geometric realisation of a graph or hypergraph is determined, up to rigid motions, by prescribed constraints such as distances or preserved normals. Classical examples include bar-joint frameworks in the Euclidean plane, where generic rigidity admits a purely combinatorial characterisation. In this talk, I will present a group-theoretic framework for rigidity developed in joint work with Klara Stokes. Using graphs of groups, we describe motions and infinitesimal motions in a unified setting encompassing a range of rigidity problems, including symmetric and surface-based frameworks. The resulting structure naturally forms a groupoid encoding compatible local deformations. I will discuss how this perspective leads to rigidity criteria generalising the classical theory in the plane, and how it relates to recent sheaf-theoretic approaches to infinitesimal rigidity on graphs. (pdf)

Strassen's famous discovery of an $O(n^{2.81})$ algorithm for general matrix multiplication became a milestone in the theory of arithmetic complexity and sparked much research. It turns out the central object is the multiplication 3-tensor, known as the structure constants of an algebra, and the arithmetic complexity is determined by the rank of this tensor. In principle the rank is algorithmically decidable, but since the trivial algorithm is an application of the Buchberger algorithm, in practice it is not. A confounding factor is that the objects considered exhibit a rather large symmetry group, which you normally would expect to make things easier, but when Gröbner bases are involved rather makes everything more difficult. Finding techniques to take advantage of symmetries is an important open problem, that perhaps SNAG could approach from new directions. I will explain the problem basics and report on some (old) work by Daniel Andrén, Klas Markström, and myself on the matter. (pdf)

I will show a generalization of Hilbert’s basis theorem to a class of noncommutative nonassociative polynomial rings known as generalized Ore extensions (GNOEs), or wildebeests. This is joint work with Masood Aryapoor. (pdf)

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group. Among graded rings, strongly graded rings form a particularly well-behaved and structurally rich class. In this talk I will introduce a notion of factor systems for strongly graded rings, consisting of algebraic data that encode both the bimodule structure of the homogeneous components and their algebraic relations. We show that strongly graded rings with fixed principal component are classified, up to isomorphism, by conjugacy classes of such factor systems. Conversely, every abstract factor system gives rise to a strongly graded ring realizing it. Furthermore, factor systems also provide a convenient framework for studying the problem of lifting derivations from the principal component to graded derivations of the whole ring. We derive explicit compatibility conditions for the existence of such lifts and interpret the resulting obstructions in cohomological terms. This is joint work with Stefan Wagner. (pdf)

Let G be a group and let R be a G-graded ring. We show that a nonzero central idempotent in R has finite support group in two broad settings: when G is abelian, and when G is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, in both settings, when G is torsion-free, every central idempotent lies in the principal component of the grading. Our results generalize those of H. Bass and R. G. Burns from group rings to noncommutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to various classes of rings. (pdf)

At SNAG 2024, I announced a class of so‑called strange algebras defined by the simple identity $S(x)=(x^3)^2−(x^2)^3=0$. The principal motivation for this class is the triviality of its Peirce polynomial, which implies that there are no a priori restrictions on the spectrum of the algebra. It is known that a fairly broad range of familiar algebras - including Jordan and medial algebras - satisfy this identity. Moreover, a related class of algebras, the palintropic algebras, was introduced earlier by Etherington. In this talk, I will present recent developments in the theory. In particular, an unexpected phenomenon has emerged: despite the vanishing of the Peirce polynomial, the fusion rules governing multiplication between Peirce subspaces can be described explicitly. A second surprising property is that, in such algebras, multiplication by an idempotent defines an algebra homomorphism. Finally, several explicit examples of S-palintropic algebras will be discussed. (pdf)

We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings. (pdf)

A standard graded commutative algebra A is called Koszul if its residue field has a linear free resolution over A. These resolutions are in general infinite, which makes proving Koszulness a challenging task. However, there are several tools one may use. The most common method is to prove that the algebra is G-quadratic, meaning that its defining ideal has a quadratic Gröbner basis, possibly after a change of coordinates. An alternative method is to find a Koszul filtration. Both being G-quadratic and having a Koszul filtration are sufficient, but not necessary, conditions for being Koszul. In our recent work, we study the relationship between these two properties. It is known that there are algebras with Koszul filtrations that are not G-quadratic. But does every G-quadratic algebra have a Koszul filtration? (pdf)

In the early eighties, Riedtmann showed that there is a close connection between the module category of a representation-finite self-injective algebra B, and that of a certain hereditary algebra A associated with B. This connection is given in terms of Auslander-Reiten (AR) quivers: The AR quiver of B can be obtained by gluing together several copies of the AR quiver of A, and taking orbits under a certain group action. A modern interpretation of her result is given by the existence of a full and dense functor from the bounded derived category of A to the stable module category of B. In this talk, I shall give an overview of Riedtmann's construction, and discuss some of the problems and possibilities that arise when trying to generalise her result from the point of view of higher-dimensional Auslander-Reiten theory. (pdf)

In this talk I will present some review and new results on n-ary Hom-algebra structures. (pdf)